Hurwitz theorem

E637304

mathematical theorem result in Diophantine approximation

Hurwitz theorem is a fundamental result in Diophantine approximation that gives an optimal bound on how well any irrational real number can be approximated by infinitely many rational numbers.

Try in SPARQL Jump to: Surface forms Statements Referenced by

Observed surface forms (1)

Surface form	Occurrences
Hurwitz's theorem	1

Statements (46)

Predicate	Object
instanceOf	mathematical theorem ⓘ result in Diophantine approximation ⓘ
appliesTo	irrational real numbers ⓘ
assumptionOnVariable	x is irrational ⓘ
boundIs	1/(sqrt(5) q^2) ⓘ
characterizes	quality of approximation of irrationals by rationals ⓘ
classification	classical theorem in number theory ⓘ
codomain	rational approximations ⓘ
concerns	approximation of real numbers by rational numbers ⓘ
conclusion	there exist infinitely many integers p and q with q > 0 satisfying \|x - p/q\| < 1/(sqrt(5) q^2) ⓘ
domain	real numbers ⓘ
extremalCase	conjugate of the golden ratio ⓘ
extremalCase	golden ratio ⓘ
field	Diophantine approximation NERFINISHED ⓘ number theory ⓘ
gives	optimal bound on rational approximation of irrationals ⓘ
hasGeneralization	Khinchin-type theorems in Diophantine approximation NERFINISHED ⓘ results on Markov numbers ⓘ
hasOptimalConstant	1/sqrt(5) ⓘ
hasProofMethod	continued fraction expansion of irrationals ⓘ properties of convergents ⓘ
implies	every irrational has infinitely many very good rational approximations ⓘ no irrational can be approximated better than order 1/q^2 with a larger uniform constant ⓘ
involvesConstant	sqrt(5) ⓘ
isPartOf	classical theory of Diophantine approximation ⓘ
namedAfter	Adolf Hurwitz NERFINISHED ⓘ
originalLanguage	German ⓘ
quantifier	for every irrational real number ⓘ infinitely many rational numbers ⓘ
refines	Dirichlet approximation theorem NERFINISHED ⓘ
relatedProblem	finding best possible constants in rational approximation inequalities ⓘ
relatedTo	Lagrange spectrum NERFINISHED ⓘ Markov spectrum NERFINISHED ⓘ best Diophantine approximations ⓘ
sharpness	bound cannot be improved for all irrationals ⓘ constant 1/sqrt(5) is best possible ⓘ
statesThat	for every irrational real number x there exist infinitely many rationals p/q with \|x - p/q\| < 1/(sqrt(5) q^2) ⓘ
topic	metric Diophantine approximation ⓘ rational approximation exponents ⓘ
typeOfBound	uniform bound valid for all irrationals ⓘ
usedIn	Diophantine analysis of quadratic irrationals ⓘ study of badly approximable numbers ⓘ theory of continued fractions ⓘ
usesConcept	continued fractions ⓘ convergents of continued fractions ⓘ
yearProved	1891 ⓘ

Referenced by (2)

Full triples — surface form annotated when it differs from this entity's canonical label.

Diophantine approximation → hasKeyResult → Hurwitz theorem ⓘ

Montel theorem → relatedTo → Hurwitz theorem ⓘ

subject surface form: Montel's theorem

this entity surface form: Hurwitz's theorem